The layer number of grids

Abstract

The peeling process is defined as follows: starting with a finite point set $X\subset {\mathbb{R}}^d$, we repeatedly remove the set of vertices of the convex hull of the current set of points. The number of peeling steps needed to completely delete the set $X$ is called the layer number of $X$. In this paper, we study the layer number of the $d$-dimensional integer grid $[n{]}^d$. We prove that for every $d\ge 1$, the layer number of $[n{]}^d$ is at least $\Omega \left(n^{\frac{2d}{d+1}}\right)$. On the other hand, we show that for every $d\ge 3$, it takes at most $O(n^{d-9/11})$ steps to fully remove $[n{]}^d$. Our approach is based on an enhancement of the method used by Har-Peled and Lidický for solving the 2-dimensional case.

7 pages